As in some other academic areas, there is a sort of conflict in math education between the rigorous teaching (through repeated practice, drilling, etc.) of basic math skills and a more learner-centered, discovery-driven style of teaching, where students are encouraged to think critically and find their own paths forward, often by inventing their own ways of doing math or by solving novel, difficult problems.

Obviously this is not an either/or proposition, and unfortunately no style of math teaching is known to result in a uniformly well-educated population. I am ignorant of actual empirical findings in this area (and in any case skeptical of many of the types of empirical findings that I suspect exist, on methodological grounds), so I do acknowledge that everything I'm saying is essentially just based on my opinions from introspection and observation of the world around me.

In order to be any good at math, in order to solve new (to you) types of problems, and so on, you do need to practice the skill of working on an unknown new problem which isn't susceptible to a specified set of skills (as opposed to most of the exercises at the end of the section of any math text, which are usually basic applications of the techniques taught in the section). I think that this type of work is also what math is, on some level, about, and that discovery-centered learning (or whatever it's called) can lead to a great appreciation of math as a field.

On the other hand, if you can't reliably add fractions (as many of my precal students cannot), your ability to explore new problems will be severely constrained. If you never learned long division because your teachers think it is boring and obsolete in the age of ubiquitous computing devices, you'll find it harder to learn polynomial long division, and when you encounter a more novel problem later, you may not even imagine it as a way forward. If your notion of a limit is only vague, you won't be able to write an analysis-style proof to solve a problem in a metrizable topological space. If you can't compute a double integral you may never understand the unique properties of the normal curve. If you can't mechnically process the symbolic logic behind a proof by contradiction, you may introduce logical errors even when you understand the argument you're trying to make.

All of which is to say that doing anything interesting at a given level usually relies on the boring techniques of previous levels.

I had a funny moment the other day when I needed to compute some zeroes of a function for my own work, because I realized I was using an exact skill I had just taught in our precal class. I think my precal students probably imagine that my work is a lot more like theirs than it actually is (almost none of my work involves computational "problems"), yet here was an elementary technique from their class which I absolutely needed to use in order to proceed.

Returning to my uninformed ramblings about younger students, there is also this. Little kids should not be bored into submission by having to do pages and pages of long division problems while being forbidden from exploring their own math interests (or discouraged from ever having those interests). At the same time, some kids (and adults) enjoy the part of math where you learn to do something neatly and properly and then execute that skill over and over. (I enjoy this aspect of proof-writing, so this enjoyment can also exist on a level well past arithmetic.) That enjoyment is not wrong or somehow antithetical to the spirit of mathematics, and the frustrations a kid may feel with never being taught a correct algorithm for doing anything and being expected to derive and explain her own methods for every new thing are also legitimate. (On a practical level, a ton of jobs are ideally suited to people who enjoy being methodical and careful, and cultivating that habit is a proper function of education.)

In my ideal world, people would have some appreciation for the abstract qualities of math, and at the same time, would feel comfortable doing the kinds of manipulations they find helpful. They'd be able to double a recipe either by using reasoning to develop an ad hoc method or by relying on a trusty algorithm for fraction multiplication. Would-be engineers would show up to college with at least the basic skills required to study calculus, and kids predisposed to be mathematicians would arrive with some experience having fun working on hard or weird problems.

3 comments:

I think once you get past a lot of the strawman versions of "rote" and "discovery-based" methods used by members of each camp to discredit the other camp, a lot of people do think that elements of both are desirable, and I think most would be thrilled beyond imagining by the kind of ideal universe you describe. Agreeing on the idea that we want people to be good at all kinds of math in all kinds of ways, and to enjoy it, is not a toughie.

I suspect that the deficient students you see in their early college career have been less harmed (or held back or whatever) by the specific type of teaching philosophy their previous math teachers used than by other aspects of their math education history. For instance, elementary school teachers who are legitimately enthusiastic about and competent in math would be a nice starting point.

Most teachers go into elementary teaching because they want to work with kids, not because they love the subject matter and want to pass it on. MANY of them understand only one way to do various types of math problems (so all the other ways are "wrong"), or don't even know how to do much math, let alone teach it, or even actually fear and hate math.

That spewing is based on my experiences in elementary education classes in a competitive school.

A co-worker just yesterday told me that she had a math teacher who liked to assign homework before teaching the concept. On purpose. This was a deliberate strategy. Which she continued for at least five years after my co-worker's dad had yelled at her for teaching idiotically when her sister had had that teacher. Maybe she thought she was using the discovery method.

Frankly, I'm for any kind of teaching that leads to students learning more about doing math than they learn about fearing or hating math.